Sklar's theorem separates the marginals of a joint distribution from its dependence structure. We apply this viewpoint to Born statistics generated by local quantum measurements. For non-product two-qubit states, the dependence nuclei over all local binary projective measurement contexts determine the state up to at most a discrete double-spin-flip ambiguity. The corresponding scalar optimization, combined with the quantum total correlation, reproduces global quantum discord.
Sklar's theorem separates the marginals of a joint distribution from its dependence structure. We apply this viewpoint to Born statistics generated by local quantum measurements. For non-product two-qubit states, the dependence nuclei over all local binary projective measurement contexts determine the state up to at most a discrete double-spin-flip ambiguity. The corresponding scalar optimization, combined with the quantum total correlation, reproduces global quantum discord.
Working with multivariate probability distributions Sklar introduced the notion of copula in 1959, which turned out to be a key concept to understand the structure of distributions of composite systems. Roughly speaking Sklar proved that a joint distribution can be represented with its marginals and a copula. The main goal of this paper is to present a quantum analogue of the notion of copula. Our main theorem states that for any state of a composite quantum system there exists a unique copula such that they are connected to each other by invertible matrices, moreover, they are both separable or both entangled. So considering copulas instead of states is a separability preserving transformation and efficiently decreases the dimension of the state space of composite systems. The method how we prove these results draws attention to the fact that theorem for states can be achieved by considering the states as quantum channels and using theorems for channels and different kind of positive maps.
Quantum discord for two-qubit X-states: A comprehensive approach inspired by classical polarization optics
Classical correlation and quantum discord are computed for two-qubit X-states. Our approach, which is inspired by the methods of classical polarization optics, is geometric in the sense that the entire analysis is tied to the correlation ellipsoid of all normalized conditional states of the A-qubit with measurement elements applied to the B-qubit. Aspects of the computation which depend on the location of the reduced state of A inside the ellipsoid get clearly separated from those which do not. Our treatment is comprehensive\,: all known results are reproduced, often more economically, and several new insights and results emerge. Detailed reexamination of the famous work of Ali, Rau, and Alber [Phys. Rev. A {\bf 81}, 042105 (2010)], in the light of ours and a counterexample we manufacture against their principal theorem points to an uncommon situation in respect of their principal result\,: their theorem turns out to be `numerically correct' in all but a very tiny region in the space of X-states, notwithstanding the fact that their proof of the theorem seems to make, in the disguise of an unusual group theoretic argument, an {\em a priori} assumption equivalent to the theorem itself.
Entangled qubits are interconnected at any separation. Quantum mechanics represents the interconnection by one unfactorizable entangled state. A measurement at one station collapses the entangled state and fixes the state assigned to the other qubit, depending on the outcome. Interpreted as physics, the entangled state and wavefunction collapse conflict with Einstein's separability and locality. We resolve the conflict by Corpuscular Quantum Mechanics, which replaces the entangled state with source constraints and reproduces the quantum joint probabilities exactly. Each qubit leaves the source carrying one of two definite states and an unshared random variable. An extended Born rule assigns a generally complex coefficient to the transition from the source-assigned state to the recorded outcome, using only what the qubit carries and the locally chosen measurement direction. Averaging products of the coefficients over the source ensemble returns the joint probabilities. Nothing is fitted. An exact identity maps the amplitudes of any pure n-qubit state to the source constraints. We prove the random variables admit a positive distribution meeting the constraints. The identity yields the joint probabilities of the Bell, Greenberger-Horne-Zeilinger, Eberhard, W, Hardy, and Dicke states for arbitrary directions. The resolution requires no action at a distance, no collapse, and no outcomes for unperformed measurements. The one change is that the event-level quantity is a transition coefficient, not a probability. Bell's theorem therefore does not apply. Quantum mechanics is thus one recipe for computing the correlations, and the entangled state is a feature of the recipe, not of nature.
We built a new set of suitable measures of correlations for bipartite quantum states based upon a recently introduced theoretical framework [Bussandri et al. in Quantum Inf. Proc. 18:57, 2019]. We applied these measures to examine the behavior of correlations in two-qubit states with maximally mixed marginals independently interacting with non-dissipative decohering environments in different dynamical scenarios of physical relevance. In order to get further insight about the physical meaning of the behavior of these correlation measures we compared our results with those obtained by means of well-known correlation measures such as quantum mutual information and quantum discord. On one hand, we found that the behaviors of total and classical correlations, as assessed by means of the measures introduced in this work, are qualitatively in agreement with the behavior displayed by quantum mutual information and the measure of classical correlations typically used to calculate quantum discord. We also found that the optimization of all the measures of classical correlations depends upon a single parameter and the optimal value of this parameter turns out to be the same in all cases. On the other hand, regarding the measures of quantum correlations used in our studies, we found that in general their behavior does not follow the standard quantum discord D . As the quantification by means of standard quantum discord and the measures of quantum correlations introduced in this work depends upon the assumption that total correlations are additive, our results indicate that this property needs a deeper and systematic study in order to gain a further understanding regarding the possibility to obtain reliable quantifiers of quantum correlations within this additive scheme.
Abstract Understanding how correlations can be used for quantum communication protocols is a central goal of quantum information science. While many authors have linked the global measures of correlations such as entanglement or discord to the performance of specific protocols, in general the latter may require only correlations between specific observables. In this work, we first introduce a general measure of correlations for two-qubit states, based on the classical mutual information between local observables. Our measure depends on the state’s purity and the symmetry in the correlation distribution, according to which we provide a classification of maximally mixed marginal states (MMMS). We discuss the complementarity relation between correlations and coherence. By focusing on a simple yet paradigmatic example, i.e. the remote state preparation protocol, we introduce a method to systematically define the proper protocol-tailored measures of the correlations. The method is based on the identification of those correlations that are relevant (useful) for the protocol. On the one hand, the approach allows the role of the symmetry of the correlation distribution to be discussed in determining the efficiency of the protocol, both for MMMS and general two-qubit quantum states, and on the other hand, it allows an optimized protocol for non-MMMS to be devised, which is more efficient with respect to the standard one. Overall, our findings clarify how the key resources in simple communication protocols are the purity of the state used and the symmetry of the correlation distribution.
Quantum correlation often refers to correlations exhibited by two or more local subsystems under a suitable measurement. These correlations are beyond the framework of classical statistics and the associated classical probability distribution. Quantum entanglement is the most well-known of such correlations and plays an important role in quantum information theory. However, there exist non-entangled states that still possess quantum correlations which cannot be described by classical statistics. One such measure that captures these non-classical correlations is discord. Here we introduce a new measure of quantum correlations which we call entropic accord that fits between entanglement and discord. It is defined as the optimised minimax mutual information of the outcome of the projective measurements between two parties. We show a strict hierarchy exists between entanglement, entropic accord and discord for two-qubit states. We study two-qubit states which shows the relationship between the three entropic quantities. In addition to revealing a class of correlations that are distinct from discord and entanglement, the entropic accord measure can be inherently more intuitive in certain contexts.